280 THEORY OF COLLINEATIONS. 
for the point A would then have to be a second invariant 
point on the line /’, which is impossible. 
THEOREM 6. The mixed group mG.( AP@) contains, in addition 
to the continuous group G.(APQ), ©? collineations of type I and 
o! of type IV, which leave A invariant and interchange P and Q. 
329. The Mixed Groups mG,(PQ) and mG,(Il’). The 
continuous group G,(P@) contains ~ collineations leaving 
Pand Q separately invariant; we seek, in addition to these, 
all collineations which interchange P and Q. There are ~* 
triangles (A A’A”’) so situated that A’A” PQ are collinear and 
the cross-ratio (A’A’”"PQ) = — 1. Each of these triangles is 
the invariant triangle of a two-parameter group in which 
collineations satisfy the relation k + k’=0, and hence inter- 
change P and Q. Therefore, there are ~* collineations of 
type I which interchange P and Q. 
Let us consider the groups G,'(A’A’’l’) of type II, where 
(A'A”PQ)=-—1. The collineations of this group depend 
upon two parameters k and t. When & = — 17, the transfor- 
mations along A’A” are involutoric and interchange P and Q. 
The group G,/( A’A’’l’) contains ~’ such collineations, one for 
eachvalueof t. The figure (A’A”’/’) can bechosen in ©? differ- 
ent positions satisfying the condition (A’A’” PQ)=—1. Hence, 
there are ~° collineations of type II which interchange P 
and Q. 
Let g be any line of the plane cutting PQ in A’ and take A”’ 
on PQ such that (A’A’’PQ) = —1. The involutoric collinea- 
tion of the group H,(A”,g) interchanges Pand Q. There 
are evidently * such involutoric collineations, one for each 
line of the plane not passing through P or Q. 
In like manner, it may be shown that the mixed group 
mG,(ll’) has a similar structure to mG,( PQ); these groups 
are dualistic, and the properties of the former may be in- 
ferred at once from those of the latter. 
